"I'm heading North, girl. Back to Winterfell," she said, "I'm finally going home. Come with me."
But Nymeria gave her a couple sniffs and then turned and ran off into the snow. Arya sat back, looking a little disappointed, but then she accepted the wolf's decision: "That's not you."
The Stark direwolves have long acted as a kind of outward representation of the Stark children's souls: when Lady was killed, Sansa lost her Stark-ness for a while; when Robb was murdered, the Freys beheaded his wolf Grey Wind; when Bran became the Three-Eyed Raven, his wolf Summer died defending him against wights; and when Ramsay Bolton tossed Shaggydog's head at Jon Snow's feet, we should have known immediately that Rickon was doomed. Ghost was Jon's companion the entire time he was in the North -- and now that Jon's journeying down to King's Landing to, perhaps, claim his rightful place as a Targaryen heir, he no longer feels that connection to his direwolf. So he gave him to Tormund, a man who looks like he'll take good care of a direwolf.
Arya lost Nymeria early when she banished her, Harry and the Hendersons-style, to protect her after Nymeria bit Joffrey in Season 1. Nymeria's been living her life independent of the rules all this time, and Arya has followed in the same mold. She's never been interested in conforming to what's expected of her, which is a good thing because it probably saved her life. With Arya and The Hound traveling to King's Landing together, echoing their previous wandering journey, they're both looking for closure on their own terms. It's what Arya's always done.
Instead of accepting a marriage proposal and becoming a high lady and ruling a castle, or traveling with Jon or Daenerys as a soldier in their army, she's setting off to finish crossing names off her list. Whether she gets to deliver the final blow to Cersei, or she helps the North win the war, or she gives assistance to The Hound in the impending Cleganebowl, one thing's for sure: That is her.